# Full text of "Treatise On Analysis Vol-Ii"

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```12   THE SPACE L°°      173

Suppose that M^/) < +00. Then for each a> M^/) the set of points
x e X such that f(x) > a is negligible. Now the set of points jc e X such
that f(x) > M00(/) is the union of the sets of jc e X such that f(x) > rn,
where (rj is a decreasing sequence with M^/) as limit. Hence (13.6.2)
we have m^f) ^f(x) ^ M^/) almost everywhere in X. It follows that
mn(f)£UJJ) if /x # 0. (If IJL = 0, then mj/) = +00 and MJ/) = -oo.)
The relation m^f) = M^/) is equivalent to saying that /is equivalent to
a constant if ^ ^ 0. Also (still assuming that ju 7^ 0) we have

inf/(x) g ess inf/(x) ^ ess sup/(x) ^ sup/(x).

xeX                 jceX                      xeX                   JceX

If two functions /, g are equivalent, then

"«(/) = ^oo (^)       and       M, (/) = M^).

Hence we can define mm(f) and M^/) for a function /which is defined only
almost everywhere in X: we choose any function g e/such that g is defined
everywhere in X, and put mm(f) = mj(g) and Mn(f) = M^g).
If /and g are such that/+ g is defined almost everywhere, then

(13.12.1)                    MJ/+ g) g MJ/) -f MUtf)

wherever the right-hand side is defined. Likewise, if /and ^ are both ^ 0,
then

with the product convention of (13.11).

A function /, defined almost everywhere on X, with values in R or C,
is said to be bounded in measure or essentially bounded (with respect to /i)
if M^d/l) < -f oo ; it follows that/is then finite almost everywhere. A bounded
function is bounded in measure.

(13.12.2) (Mean value theorem) Let f '• X->R be measurable and
bounded in measure. For every integrable function g ^ 0, the function fg
(which is defined and finite almost everywhere) is integrable, and

(13.12.2.1)           mj/) (g d» £ (fg d^ ^ MM) \9 dp*

Furthermore •, two of the three terms in (13.12.2.1) are equal only if, in the
(measurable) set S of points xe X such that g(x) ^ 0, we have either f(x) =
MO^/) almost everywhere, or f(x) = m^f) almost everywhere.) = -MJ-/).h satisfy the
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